On quaternionic line bundles
by
Gustavo Granja
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 1999
@Gustavo Granja, MCMXCIX. All rights reserved.
The author hereby grants to MIT permission to reproduce and to distribute publicly
paper and electronic copies of this thesis document in whole or part and to grant
others the right to do so.
Author.....
. .
.
. . . . . . . . . . . . . . ....
..(.
Department of Mathematics
May 7, 1999
Certified by..
... .. .. .. .. ... .. .. ... .. .. .... .
....
Michael Hopkins
Professor of Mathematics
Thesis Supervisor
Accepted by ...............
S
...
..
...
....................
.
...
Richard Melrose
Chairman, Departmental Committee on Graduate Students
i
''HI
TS
ITTT'
TFCHNOLOGY
L A'
On quaternionic line bundles
by
Gustavo Granja
Submitted to the Department of Mathematics on May 7, 1999
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
Abstract
We use obstruction theory based on the unstable Adams spectral sequence for S 3 to study the
classification of quaternionic line bundles over finite dimensional complexes. We pay special attention
to universal examples, namely finite dimensional quaternionic projective spaces.
Thesis Supervisor: Michael Hopkins
Title: Professor of Mathematics
Acknowledgements
I am very grateful to Mike Hopkins for the patience he has shown toward me during the three
odd years that he served as my advisor. I am also indebted to him for suggesting the use of the
unstable Adams spectral sequence and I would like to thank both him and Haynes Miller for the
courses they taught during my stay at MIT.
I owe most of my understanding of algebraic topology to Dan Dugger and Nitu Kitchloo. During
our final year here, my almost daily talks with Dan must surely have tried his patience, but I left
every single one of them reminded of how much fun Math can be. Usually, I really needed to be
reminded.
Many thanks also to Sharon Hollander (who will make a great cult leader some day), Jesper
Grodal, Bobby Kleinberg, Ioanid Rosu and Andrei Radulescu-Banu for many interesting discussions
and excellent talks in the Baby Topology Seminar.
Among the people who do research in less captivating areas of Mathematics, I would like to
thank Phil Bradley, Sergiu Moroianu, Carmen Young and Colin Ingalls for their friendship and
many interesting mathematical discussions. Phil and Sergiu, in particular, were my officemates
during this last year and somehow managed to resist shooting me.
My parents, my brother and Miguel Hernando certainly deserve their own paragraph for their
continued support which is very much appreciated.
Finally, I am happy to acknowledge the financial support of Programa Praxis XXI da Fundagio
para a Ci6ncia e Tecnologia, Fundagdo Calouste Gulbenkian, Fundagio Luso-Americana para o
Desenvolvimento and MIT. I am even happier that it was generous enough to provide for much
needed recreational drugs.
7
1. INTRODUCTION.
Let H denote the division algebra of quaternions. A quaternionic line bundle is a fibre bundle
with fibre H and structure group HX acting on H by left multiplication. The group of quaternions of
unit length is canonically isomorphic to SU(2) so there is a 1-1 correspondence between quaternionic
line bundles and principal SU(2)-bundles.
Unlike real or complex line bundles which are classified by their characteristic classes, quaternionic
line bundles are not. As the group of unit quaternions is nonabelian, the classifying space for
quaternionic line bundles HPI is not an Eilenberg-Maclane space. This makes the problem of
classifying quaternionic line bundles over a space X very hard. Consider for example the case when
X is a sphere. The homotopy long exact sequence of the universal bundle
S'
-+
ES3 -- * HP*
shows that 7riHP 0 = i S3 so classifying quaternionic line bundles over spheres amounts to
describing the underlying sets of the homotopy groups of S3, a problem whose solution is nowhere
in sight.
The previous example illustrates the close relationship between the classification of quaternionic
3
3
line bundles and the structure of the homotopy groups of S . Although the structure of 7r.S is
far from being completely understood, much information about it is known. This is best expressed
in terms of the unstable Adams spectral sequence. What we will do in this thesis is to apply the
available information on 7r.S 3 to the classification of quaternionic line bundles.
The first result which we will prove is a sufficient condition for a cohomology class in H 4 (X; Z)
to be the Pontryagin class of a quaternionic line bundle over a finite dimensional CW-complex X.
4
We will find N(d) such that if X is a CW-complex of dimension < d and a E H (X; Z) is divisible
by N(d) then a is the Pontryagin class of a quaternionic line bundle over X. See Proposition 3.6
for a precise statement.
Most of this thesis deals with classification of quaternionic line bundles over HP'. A quaternionic
- HP' or equivalently, by cellular
line bundle over HP' is a homotopy class of maps HP'
approximation, a homotopy class of self maps of IHP'. In this case, the classification is of special
interest for two reasons. The first is that finite dimensional quaternionic projective spaces play a
universal role: a self map of HP' will give an operation on quaternionic line bundles over complexes
of dimensions < 4n. The second, which was the original motivation for this work, is that the study
of self maps of HP' gives information about the homotopy properties of the standard multiplication
of the Lie group SU(2).
Stasheff [St] shows that there is a hierarchy on the set of homotopy classes of self maps of a loop
space X
[X, X] = A 1 (X) D A 2 (X) D ... - A,(X)
where denoting the loop multiplication on X by I,
A 2 (X) = {f E [X, X] : f o p ~ I o (f x f)}
is the set of H-self maps,
A 3 (X) = {f E A 2 (X) :
(o
(f x f) x f) ~ p o (f x A o (fx f))}
and in general An(X) is the set of H-self maps of X which "preserve associativity up to (n - 2)th higher order homotopy" (see [St]). Ao(X) is the set of loop self maps. When X = SU(2),
A 1 (X) = Z and results of Stasheff [St] imply that there is a self map of HP of degree k (i.e. whose
restriction to HP' has degree k) if and only if k E An(SU(2)). Thus the homological classification
of self maps of HPn is the same as the determination of the sets An(SU(2)).
Finally, knowledge of the sets An(SU(2)) would be of interest because it would give an indication
of which power maps of a finite loop space can be expected to admit an An-structure (cf. the work
of McGibbon [MG2, MG3] for n = 2,3).
8
Our main result is the construction of a number of new self maps of HP' (see Theorem 5.10
for a detailed statement). The main method used in the proof will be obstruction theory based on
the unstable Adams spectral sequence for S3 . As a corollary, a conjecture of Feder and Gitler [FG]
regarding the homological classification of self maps of HP' is verified for n < 5.
Organization of the paper. In section 2 we record the information about the unstable Adams
spectral sequence for HP' that we will need in the rest of the paper. In section 3 we use this to
prove the result on the existence of quaternionic line bundles over finite dimensional CW-complexes.
The remaining sections are concerned with self maps of HP". In section 4 we define the obstruction to extension of a self map of HP' and prove some of its basic properties. We also discuss the
homotopy classification of self maps and obtain it in low dimensions in terms of the homological
classification. The latter is the subject of section 5 where we construct quaternionic line bundles
over HP' with specified Pontryagin classes and as a corollary prove the Feder-Gitler conjecture for
n < 5. Section 6 is devoted to some remarks on certain interesting spherical fibrations over HPn.
9
2. THE UNSTABLE ADAMS SPECTRAL SEQUENCE FOR
HPP.
In this section we record some facts about the unstable Adams spectral sequence for HPO and
explain how to use the spectral sequence as the basis of an obstruction theory for constructing maps
into HPIP . A good general reference for the undefined terms we will use concerning the Steenrod
algebra is [Sc].
The Massey-Peterson spectral sequence. Let U denote the category of unstable modules over
the mod p Steenrod algebra. If M E U, U(M) denotes the free unstable algebra generated by M
and for P E U a free module, we write K(P) for the mod p generalized Eilenberg-Maclane space
such that
H*(K(P); Z /p) = U(P)
Recall from [HM2] that if X is a simply connected space such that
H*(X; Z/p) = U(M)
for some M E U, then the Massey-Peterson spectral sequence is defined. It is the spectral sequence
of a tower of principal fibrations under X
(2.1)
E2
El
X
>Eo = K( Po )
K(Q2p
> K(QP 2)
> K ( P1
where
0 +- M <-
PO +- P 1 +- P 2 --
..
is a free resolution of M. The diagram (2.1) is called an unstable Adams resolution for X. For a
simply connected space of finite type this spectral sequence converges strongly and
E2'(X) = Extst(M,F2 )
f
=> 7rt-X
0
2
Let I(A) denote the augmentation ideal of the Steenrod algebra. Recall that if M, N E U, a map
: M -+ N is minimal if ker(f) C I(A)M. A resolution is minimal if each of its maps is minimal.
Lemma 2.1. If H*(X) = U(M) there is an Adams resolution of X so that in the corresponding
homotopy spectral sequence
Efs't)(X) = E.',
(X)
Proof. Pick a minimal resolution P. for M. This has the property that all the differentials in the
complex Homu(P., VZ/2) are trivial (see for example [MT]). Since this complex is the El-term
of the Massey-Peterson spectral sequence we conclude that, as bigraded vector spaces, E 2 (X) =
E 1 (X).
10
We will consider the associated tower of principal fibrations over X obtained by setting X8+' to be
the homotopy fiber of X -+ E,.
(2.2)
P2
2
>K(Q2p)
PI
1
>K(fP 1 )
IPO
j
X0 -- X
>
K0(Po)
The homotopy spectral sequence of this tower is clearly isomorphic to the Massey-Peterson spectral
sequence.
Definition 2.2.
through X 8 .
The Adams filtration of a map
f :W
-+
X is the largest s such that f factors
We can use the tower (2.2) to construct maps W -+ X as follows. Assuming by induction that
we have constructed a map on the k-skeleton fk : W(k) -+ X of Adams filtration s, it follows
that the obstructions to extending the map to the (k + 1)-skeleton W(k+1) have filtration > s.
This restricts the set of obstructions we have to rule out in order to guarantee that fk extends to
fk+1 : W(k+1) -+ X.
In order to proceed by induction we will need an estimate of the Adams
filtration of the extension fk+1. The following lemma will be useful for this purpose. It gives a
criterion for the vanishing of the obstruction to extension to imply the vanishing of the obstruction
on an Adams cover. If s > t, let p' = pt o
o. p,-.
Lemma 2.3. Suppose that in the homotopy spectral sequence of (2.2) the image of d, in E,+it+i
8
is 0 for each r > q and i > 0. Then p'. is injective on pq_
8 Xs+q-1) C 7rt- 8 X .
Proof. Write F, = K(Q'P,) and consider the (q - 1)-th derived exact couple of the homotopy exact
couple of (2.2):
q-s+1F
X_
6
(q-1)
(q 1)
PS-1
7r-+F(q-1)
1
6
(q-1)
l7rtsXs-2
Ps-2
Fst3+
__X_- rtF
I
By definition
rtsXs(q-1)
7t__,Xs*-l
_
7t_,Xo(--1))
=
p
8-
7
is nonzero.
F (q-1)
-F
Ps-3
t_(
8 X+q-1)
c
7rtXs.
Suppose a
E ker(p',0
11
Then there exists u < s and 0 5 y E im(6) C irt-,Xu(q-1) such that pus.(a) = y. Since
holimX 8 e * there also is v > s and 3 E 7rt-,X"(-1) such that p (,#) = a and 3 ( imp,*. This
gives a nonzero differential d, : Eu,t+u8+l
{Z/p
The spectral sequence for
HP.
-+
E,.,t+v-8 and concludes the proof.
Let M denote the following unstable module
< X, P 2 x, P2p p2 X, P2 p2p p2 X,... >
Z/p < x, Sq4 x, Sq Sq4 x, Sq 6 Sq Sq4 x,... >
if p > 2
if p = 2
where x denotes an element of degree 4. Then H*(HP"; Z/p) = U(M). In general, Adams resolutions are not preserved by looping and so the unstable Adams spectral sequences of X and QX are
quite different. In our special case, however, we have the following
Lemma 2.4. The Massey-Peterson spectral sequences for S' and HP'
For each r > 2, s, t
E,.t',(S
3
are naturally isomorphic.
) = E,st+l(HPP)
3
Proof. First note that H*(S 3 ; Z/p) = U(E3Z/p) and QM = E Z/p. Since Q 1 M = 0, applying Q to
3
3
a free resolution of M yields a free resolution of E Z/p. Thus an Adams resolution for S can be
L
obtained by looping an Adams resolution for HP' which implies the result
3
The previous lemma makes things easier for us since the Adams spectral sequence for S has been
studied extensively. As usual, if p is an odd prime, we set q = 2(p - 1).
Figures 1 and 2 describe the portion of the E 2 term of the unstable Adams spectral sequence for
IHP1 along the vanishing line. As usual, vertical lines represent multiplication by p and the slanted
lines composition on the right with qj if the prime is 2 and with a, if the prime is odd.
Theorem 2.5 (Mahowald, Miller, Harper-Miller, Thompson). Consider Figures 1 and 2.
(a) Above the classes shown and the dotted lines in the columns where no classes appear, the E 2
term vanishes.
(b) The classes in dimensions qk + 3 for p odd and the circled classes in dimensions 8k + 7 and
3
filtration > 4k + 2 for p = 2 correspond to elements on 7r. S which are stable and detected by
the real e-invariant.
(c) The classes in dimensions qk + 2 for p odd are not boundaries.
Proof. Recall from [Mal] and [HM1] that for each prime there is a bigraded complex (A(3), d) such
that
3
E2' (S )
= Hst-3A(3)
There is a short exact sequence of complexes
0 -- * A(1) -+ A(3) -+ W(1) -
0
which induces a split short exact sequence on homology. A(1) is a complex with 0 differential which
corresponds to the Z-tower in t - s = 3.
Let M 0 A denote the E 1-term of the stable Adams spectral sequence for the modp Moore
spectrum given by the A-algebra. The main results of [Mal] and [HM1] state that there is a map of
complexes W(1) -+ M 0 A inducing an isomorphism on homology for t - s < 5s - 16 for p = 2 and
t - s < (p + 1)qs - (p + 2)q for p odd. This reduces (a) to a statement about the stable E 2 -term of
the Moore spectrum except in low dimensions where it is easily checked directly by computing the
homology of the complex A(3).
The E 2-term of the Moore spectrum is computed in the required range in [Mi] for p odd and in
[Ma3] for p = 2. It agrees with the description given in Figures 1 and 2 shifted according to Lemma
2.4. This proves (a).
12
s
5
4
3
2
- "
11
0
4
5
6
7 8 9 1011
t-
s = 4k+
3
2
1
0
-1
-2
t-s=8k+
4 5 6
7
8 9 1011
FIGURE 1. E"'t)(IHIP') for p = 2.
s
3
-0
4
s
3+q
2 +2q
t -s
k+
0
1
[
-1
t - s=qk+
2 3
FIGURE 2.
2+q
E s't)(HPoP) for p odd.
13
Let M" denote the modp Moore space with top cell in dimension n and i : S'-1 -* Mn denote
the inclusion of the bottom cell. Let A: Mn+r -+ Mn with r = 8 if p = 2 and r = q if p > 2 be a
map inducing an isomorphism in K-theory (see [Ad]).
2
3
Suppose first that p is odd. Let d denote an extension of a generator a E 7r2pS to M P+l. Then
3
by [Ad, Proposition 12.7], ak = d o Ak-1 o i E 7rqk+2S survives to a stable class which is detected
3
by the e-invariant. Since ak has Adams filtration at least k it must be represented in E 2 (S ) by the
class in bidegree (t - s, s) = (qk + 2, k).
Next let p = 2. Let ft denote an extension to M 13 of the generator p of 7r12 S 3 . Then pk =
ft o Ak-1 o i E 7r8k+4S 3 has Adams filtration at least 4k + 1 (the Adams filtration of the Adams
3
map is 4 since we are in the stable range). It follows that the Toda bracket < Pk, 2, 77 >E 7r8k+7S
has Adams filtration at least 4k + 2. Proposition 12.18 of [Ad] shows that this bracket survives to
a stable class of order 4 detected by the e-invariant so the bracket is represented on E2 (S3) by the
class in bidegree (t - s, s) = (8k + 6,4k + 2). This completes the proof of (b).
3
4
4
3
Finally, let # denote an extension of a 2 E 7r4,+3S to M p+ . Then Ok = ,o Ak-1 o E rq(k+1)+1S
0
has Adams filtration at least k + 1 and is not null by Theorem 1.3 of [Th]. This proves (c).
Remark 2.6. In the previous theorem we have only stated the results we will need in the sequel.
The classes shown in Figures 1 and 2 are all permanent cycles and represent the vi-periodic homotopy
of S 3 . The circled classes are stable and the filled ones are killed by double suspension. For more
details, see [Ma2] and [Th].
The previous theorem gives partial information on all the homotopy groups of EHP . We will
also require complete information in low dimensions. Given Lemma 2.4, the following theorem is a
small part of the main result of [CM].
Theorem 2.7 (Curtis and Mahowald). The Adams spectral sequence for HP'
t - s < 25 is completely described by Figure 3.
at the prime 2 for
For an odd prime, we have the following result
Proposition 2.8. Let p be an odd prime. Then E2't(HP ) is described by Figure 4 for t - s <
( 2p+ 1)q - 1.
Proof. By Lemma 2.4 we can consider E2't(S3) instead. As was already mentioned in the proof of
Theorem 2.5, it follows from [HM1] that if M denotes the modp Moore spectrum (with top cell in
dimension 1), there is a map
E28'1(S3)
E 's -1,-q-3 (m)
defined for t > s > 0 which is an isomorphism for s > 3 and 0 < t - s < (2p + 1)q - 2 and an
epimorphism for s > 2 and t - s < (2p + 1)q - 2.
t
By the Localization Theorem [Mi], for s > 3 and t - s < 2pq - 4, E2' (M) is isomorphic to
FP[vI1 , bi,o] 0 E(hi,o, h 2 ,o)
where the bidegrees (t - s,s) are given by lvii = (q, 1), 1hi,oI = (2(p' - 1) - 1,1), and Ib(1,o)I =
(2p(p - 1) - 2,2).
Thus, in order to prove the proposition, we must check that for s < 3 and t - s < (2p + 1)q - 2,
E2'-(S3) = 0 except for one class in each of the bidegrees (t - s, s)
(kq+2,k)
(kq+1,k)
(3, (p + 1)q)
(3, (p + 3)q + 2)
(3, (p + 3)q+ 1)
1<k<3
1<k<3
14
S
15
10
5
0
0
4
7
FIGURE
11
15
3. E2f'(HPP)
19
at p
=
23
t-s
25
2 for t - s < 25.
s
9
0
6,
4
34
4
0
4
q+3
0
0
(p+1)q+1
2q+3
FIGURE 4. E2't(HP
(p+3)q+1
(2p+1)q-1
t-s
) for t - s < (2p + 1)q - 1, for p odd.
We will show this by using the A-algebra. Tangora's memoir [Ta] is a good reference for any
terms we use which are unfamiliar to the reader. See also [HM1]. Recall that the vector space
A(3) C A(3) generated by the set of admissible monomials with A-endings is a subcomplex and the
inclusion induces an isomorphism in homology for t - s > 0.
A basis for A(3) for s < 3 is given by
s=1
A,
s=2
A1A-(1<m<p)
/,iAm(1 < M < P)
s=3
AlAmAk(1
<< p, 1
k
Mp)
<M < p, 1
k
Mp)
k
k
Mp)
Mp)
IAmAk(1
AlpmAk(1 < M < p, 1
/.LitmAk (1 < M < p, 1
15
To compute the homology of A(3) it is useful to notice that the differential preserves the Cartan
degree (i.e. the number of p's in a monomial).
A1 ,A1 A1 and p1 A1 are cycles. For m > 1, the leading term of d(A1Am) is -mAiAm-ijA so the
differential is injective on the subspace generated by A Am with 2 < m < p. Similarly one checks
that d is injective in the subspace generated by p1Am for 2 < m <p. This computes E,'t(S3) for
s <2.
For s = 3, we need only consider the monomials for which m + k < 2p since the others lie outside
the range we are interested in. We have a lower bound on the dimension of ker d given by the rank of
the image of d plus the number of classes in filtration 2 on the Moore spectrum. We can get a lower
bound for the rank of d by giving xi E E 3,* such that the leading terms of {dxj} are all distinct.
We then must show that the two bounds agree. We will give the details only for the Cartan degree
2 summand in A(3) since this is the only case of interest for Corollary 2.9.
First note that piptiA
is a cycle. The leading term of d(plpmAk) is
-kp1pmAk-1Ai
(1-m)pipm1AAj
p1Ampp-ijA
if
1<k<pork>p
if
if
k=1 11<m<p
k=p
so we see that the only monomials of Cartan degree 2 whose differentials have the same leading term
are p1 pIA 2 and p 1 iip.+ 1A for 1 < 1 < p.
If p 0 1 mod 4, replace pip1 A2 with
p1ipA2 + 2 pjL+1k
The leading term of the differential of this element is now piiA
1 which shows that d is injective
on the linear span of {pipmAk : m or k $ 1} in the range under consideration.
If p = 1 mod 4 replace pi1 iA2 with
1-i
2
pipjA2 + 2 P1pj+1kj + 13
jt~A
The leading term of the differential of this element is p1pAjlt
This completes the proof.
3
Corollary 2.9. Let p be an odd prime. Then r4 n_ 1 (S )
&Z
2 Aj
which again shows d is injective.
0
is detected by the e-invariantfor
n < (2p + 1)q/4 but not for n = (2p + 1)q/4.
Proof. The first part follows immediately from Proposition 2.8 and Theorem 2.5 (b). Let y denote a
2
2
( 3). Then y is a permanent cycle because, for p > 3 there are no classes in
generator of E44P+l)25
the A-algebra ElP+lq 3 (S ) and for p = 3 the only class is the generator of E2' which is not a
boundary by Theorem 2.5. Since Es,(2+l*-1 = 0 for s < 2 we conclude that y detects a nontrivial
homotopy class in 7(2p+1)q-2S3 . This class is not detected by the e-invariant. Otherwise it would
be vi-periodic and it follows from Corollary 1.2 of [Th] that there is only one vi-periodic class in
L
this dimension, namely the one accounted for in Theorem 2.5.
16
17
3.
PONTRYAGIN CLASSES OF QUATERNIONIC LINE BUNDLES.
Let X be a finite dimensional CW complex. In this section we give sufficient conditions for
elements of H"(X; Z) to be Pontryagin classes of quaternionic line bundles.
We will write P for the universal Pontryagin class, which is a generator of H"(HP ; Z)
11P 0
-
K(Z,4)
P is a rational equivalence so we would expect that given a E H"(X; Z), some multiple of a can be
realized as a Pontryagin class. This is in fact the case as we will now see.
Definition 3.1. Let k E N. We say that d E Z is a k-divisor if for all CW complexes X of
dimension < k and a E H4(X; Z) such that dia, a is the Pontryagin class of a quaternionic line
bundle over X.
We denote the set of k-divisors by D(k).
Let t4 denote the fundamental class in H4(K(Z, 4); Z). If X is a CW-complex, X(k) denotes the
k skeleton of some cellular decomposition. In all that follows, the choice of cellular decomposition
will not play a role.
Lemma 3.2. d E D(k) if and only if there is a bundle over K(Z,4)(k) with Pontryagin class dt 4.
Proof. The condition is obviously necessary. Conversely, suppose a bundle
K (Z, 4) (k) _4 Hpoo
with Pontryagin class dt 4 exists and X has dimension < k. By cellular approximation, any class
o E H4(X; Z) factors through K(Z, 4)(k)
K(Z, 4 )(k)
X
and clearly
(
-
K(Z,4)
o 3 classifies a bundle with Pontryagin class do.
0
Because the rationalization of HPI is so simple, it is easy to assemble maps to HP from local
maps. Let X(p) denote the p-localization of the nilpotent space X and X(o) denote the rationalization.
Recall from [Su] that X is the homotopy inverse limit of the diagram
(3.1)
X(2 )
X(3)
X(5 )
X(o)
where all the maps are rationalization maps. If X = HPI they classify the fundamental classes
HP(' -+ K (Q, 4)
We will abuse notation and use the same letter to denote an element in H4(X; Z) and its image
in H4 (X; Z(p)) under extension of coefficients.
Lemma 3.3. Let X be a space. Then a E H4(X;
bundle over X if and only if a is in the image of
[X, HP(f)or
for every prime p.
Z) is the Pontryagin class of a quaternionic line
[X, K(Z(), 4)
18
Proof. The condition is clearly necessary. If ( ()) are local bundles with P.( (p) = a then their
rationalizations agree up to homotopy. By (3.1), a choice of homotopy yields a bundle
with
Pontryagin class a E H 4 (X; Z).
L
Fix a prime p. Definition 3.1 and Lemma 3.2 have the obvious local formulations: d E Z(,) is a
p-local k-divisor if for all CW complexes of dimension < k, dH 4 (X; Z(p)) is in the image of P. and
we denote the set of p-local k-divisors by Dp(k). As before, d is a p-local k-divisor iff there exists
K(Z(p),4)(k)
+ Hpoo
(p)
with P,(() = dt4 .
Proposition 3.4. There exists ap(k) E Z () such that Dp(k) = pap (k)Z ().
Proof. For any 1 E Z(p) there is a self map of K(Z(p), 4 )(k) inducing multiplication by I on H 4 (-; Z(p)).
This gives an action of Z(p) on the set of p-local k-divisors for every k. Therefore we can take
ap(k) = min v,(l)
1EDp(k)
where vp denotes the p-adic valuation.
I
Note that the functions k -4 ap(k) are nondecreasing. Moreover, since P : HP
-K(Z(p), 4) is
a p-local equivalence through dimension 2p + 1, ap (k) = 0 for k < 2p + 1. In particular the product
appearing in the following statement is finite.
Corollary 3.5. D(k) =
l,
D,(k) = (]ipPa (k)) Z
Proof. It follows from Lemma 3.3 that
D(k) -n
Dp(k)
P
On the other hand, if d E D(k) let
the diagram
be a bundle over K(Z, 4 )(k) with Pontryagin class dt 4 . Consider
(K (Z, 4) (k))
P
K (Z, 4)(r
K (Z (p), 4)
k
The map J(p) is a k-equivalence, so the inclusion i of the k-skeleton of K(Z(p), 4) lifts and the
composite
K (Z (p), 4) (k) -_+ (K (Z, 4) (k))()-4HP(Oo
shows that d E Dp(k) which completes the proof.
0
We would like to determine the functions ap(k). It is not hard to find an upper bound for ap(k)
just using elementary obstruction theory. Fix a prime p. Let HPI [r] denote the homotopy fiber of
a generator of H 4 (HP() ;
p)
HP('P) --- + K (Z/lp', 4)
Then, pr is a k-divisor if and only if the map
HP' [r] -+ K (Z(p), 4)
classifying the fundamental class in H 4 (HP, [r]; Z(p)) admits a section over the k-skeleton. The
obstructions to extending a section lie in
H" (K(Z (p), 4); 7r,,-1EPO < 4 >)
19
At an odd prime, p is an exponent for ir.(HP < 4 >) and at the prime 2, 4 is an exponent [Se].
Since the obstructions are natural, the pullback square
> HPr]
PO [r + 1]
K(Z(p), 4)
P > K(Z(p), 4)
a 2 (k) + 2.
shows that ap(k + 1) 5 a(k) + 1 for p odd and a 2 (k + 1)
provides much better bounds
sequence
spectral
Adams
the
on
based
theory
However, obstruction
for ap(k). Let
5
0
if k
i
4i+j+1
4i + 4
4i + 5
if k=8i+5+jwith 1 <j
if k=8i+9
if k=8i+9+jwith 1<j
<
3or j=Oand i>O
3
and for p odd,
(0
" i+1
if k <3+q
if k=iq+j with 4<-j :52+qorj =3and i > 1
These functions describe the vanishing line in the E 2 term of the unstable Adams spectral sequence
for HP'.
Proposition 3.6. ap(k) < O3(k).
Proof. We may assume k > 4. We need to show the existence of a map K(Z(p),4)(k) - HPg
which is multiplication by pOp3(k) on 7r4(-). Consider an Adams resolution of HPgy as in Lemma
2.1. Then, using the notation of (2.2), r 4 (X" ) = Z(,) and the maps
Xm
"+ X"
induce multiplication by p on 7r 4 (-).
Hence it is enough to produce a map
K (Z(p, 4)(k)
_
XOp(k)
4
C K (Z (), 4) (5)
inducing an isomorphism on 7r4(-). We certainly have such a map on S
Since in the homotopy spectral sequence of (2.2) El = E 2 , the vanishing line for the E 2 term of the
Adams spectral sequence described in Theorem 2.5 implies that the space XQp(k) is (k- 1)-connected.
E
Therefore the obstructions to extending the map from S4) to K(Z(p), 4 )(k) vanish.
A lower bound for ac(k) can be obtained by computing the image of the Pontryagin class map
on particular spaces. For instance the results of section 5 (see Proposition 5.5) imply logarithmic
lower bounds for ap(k). However, low dimensional calculations indicate that the upper bound given
in Proposition 3.6 is close to the actual value of ap(k).
20
21
4. HOMOTOPY CLASSIFICATION OVER HPn.
In this section we discuss the classification of quaternionic line bundles over
the classification for n < 3 in terms of the homological classification.
First note that by cellular approximation we have
[HPn, IP"
HP" and we obtain
-~-i+ [Ppn, ifp"]
so the classification of line bundles is the same as the classification of self maps of HP". It is more
convenient to take the latter point of view to describe the homotopy classification. Again by cellular
approximation, we have a chain of restriction maps
+ [Hp,, Hp,]
Z = [Elpl, HpI ] +- [Hp2, Hp2] -...
Definition 4.1. Let
to HP'.
f
be a self map of HP". The degree of f is the degree of the restriction of
f
Thus we see that the classification problem breaks up into two steps:
(i) Determine which self maps of HPn extend.
(ii) For each map which extends, classify the possible extensions.
We'll begin by dealing with (i). Recall that we have the Hopf fibration
S 3 -4 s4n+3
_h+
ppn
Since i is null
~ s 3 x Qsan+3
Q1HP
and an explicit homotopy equivalence is given by the product of the two maps
S3 E+ QS4 -- + QHPn
and
QS4n+3 Qh fHpn
so that we have an isomorphism
7r,(S
4
n+ 3 ) 0r,_1S3
D
- 7r(HPn)
(a, 3)
Definition 4.2. Let f : HPn
-+
'-*
h,(a) + E.(3)
HPn be a map of degree k. Then o(f) E
7r4n +3HP"
is defined by
o(f) := f.(h) - kn+1h.
Proposition 4.3. Let f be a self map of HPn. Then
Moreover o(f) E E.(7r 4 n+ 2 S 3 ).
f extends to HPn+l if and only if o(f)
= 0.
Proof. Consider the diagram where the lines are cofiber sequences
h
S4n+3
:
fp"
ffpn+1
l:
f
I
p"
> H~n+1
i
s4n+ 3
h
If the dotted arrow exists, it follows that 1 = kn+1. Since the lines are also fiber sequences through
dimension 4n + 5, the extension exists if and only if f.(h) - kn+1h E 7r 4 n+ 3 (IHP") vanishes. It
remains to see that f. (h) - kn+1 h is in the torsion summand of 7r4n+3(HP). This follows from the
E
fact that HP = K(Q, 4) and so the obstruction is 0 rationally.
Remark 4.4. Note that equivalently we can define the obstruction to extension as the composite
s4n+3 _ hjppn --
,
yjPn -- + 1ffp*
22
Now, to handle (ii), let f be a self map of HP" such that o(f) = 0.
determined by a choice of nulhomotopy of the composite
s4n+ 3
Hp"
_h+
_-+ Hp
-_+
An extension of
f
is
Hpn+i
These nulhomotopies form a set on which w4n+4 (HPn+l) = 7r4n+3(S 3 ) acts transitively. To describe
the action of this group more explicitly, let
V : HP"
4
_+ HP" V S "
be a cellular approximation of the map
1[n
idxcl
X
S4n
where c : HP -+ S4n is the map collapsing the 4(n - 1)-skeleton. Then V sends a generator of
H 4 n (HPn; Z) to the sum of the generators of H 4 n(HP; Z) and H 4 n (S 4 n; Z).
Definition 4.5. Given a
E w4n+ 4 Hpn+1 and g a self map of HPn+1 the coaction of a on g is the
composite
g
H a: Hpn+l
-Y*
HPn+1 V S4n+ 4
L4 jpn+l
The action of W4n+4 (HPn+,) on the set of extensions of f parametrizes the set of homotopy
classes of extensions relative to the 4n-skeleton. 7r 1 (Map(HPn, lHP)f) acts on this set and the
orbits of this action form the actual homotopy classes of extensions. Letting Ext(f) denote the set
of extensions of f and picking an identification of the set of relative homotopy classes of extensions
with 7r4n+3(S3 ) we have an exact sequence of sets
(4.1)
7r 1 Map(HPn, HPn)
4 w4n+ 3 (S 3 ) -+ Ext(f)
To calculate these exact sequences for n < 3 we will need some formulas for the obstruction to
extension.
Proposition 4.6. Let
f, g be self maps of HP" of degree k and 1 respectively. Let a E 7r4n (HPn).
Then
(a) o(f H a) = o(f) ± a o (nh) + k[t 4 , a]
(b) o(f o g) = Ino(f) + ko(g)
Proof. Consider the diagram
1H1Pfv S4 "
s 4n+ 3
h
h
fPIpn
f-cr
>
HJPn
Recall from [Wh] that
(4.2)
7r4n+ 3 (HiP
V S 4") = 7r4n+ 3 (HP") E 7r4n + 3 (S4)
D 7r4 (HP") 0 7r4n(S 4")
with the last summand embedded by the Whitehead product of the fundamental classes. The
projections onto the first two summands in (4.2) are given by composition with the collapse maps
on to the wedge summands. Let X denote the cofiber of
s4n+3 -,'+ HP" V S4"
4
Let x be a generator of H (X; Z), z a generator of H 4 n+ 4 (X; Z) and y an element in H 4 n(X; Z)
restricting to the generator of S4n. Then if xy = kz, the component of / along the third summand
is kk[t 4,t 4 n].
Since by [Ja], c o h = nV4n we conclude that in terms of the decomposition (4.2), we have
V o h = h nv4+ [t4, t4n]. Hence we get
o(f H a) = o(f) ±a o (nh) + k[t 4 , a]
23
which proves (a).
As for (b)
o(f o g)
=
=
=
f,(g,(h)) - (kl)"h
f,(lnh+o(g)) - (kl)nh
l"(knh + o(f)) + ko(g) - (kl)nh
as required.
We can also obtain a formula relating composition with the coaction
Proposition 4.7. Let
Then
f, g be self maps of HPn of degrees k, 1 respectively and a,3 E ir4nHPn.
(g--
) o (f - a) = (g o f) -A (k"f3+la)
Proof. Note that we have a (4n + 3)-equivalence
HPn V S~ -4n+3 HP
x
so denoting by kn : HP" -+ S4" the composition of the degree kI map with the collapse of the
4(n - 1)-skeleton of HP", we have a commutative diagram
1HPn V s 4 n
Iv
V">
Vk
k4
h3
hence
(gV/)o(f xk") = (gof) -k"n3
By restricting to each of the wedge summands we see that the following diagram commutes
HPn
v
s4n
(f xk )V(axO) >
V
fqlp
Hp
HPn
v
-,
s4n
gV/3
We conclude that
(g VI) o ((f
x
k") V (a
x
0)) o
=
(((g o f) - kn) V la) o V
=
((g o f) -i k"3) - la
=
(go f)-I(kn3 + la)
E
as required.
We can now describe the homotopy classification of self maps of HP" in terms of the homological
classification for n < 3. Let S(n, k) be the set of self maps of HPn of degree k.
The order of the sets S(n, k) for k odd and n < 3 has also been obtained by Iwase, Maruyama
and Oka in [IMO] using similar methods.
Proposition 4.8. Let S(n, k) be the set of self maps of HPn of degree k.
nonempty. Then
(a)
(b)
IS(2, k)
S(3, k) =4
2
1
if k is odd
if k is even
2
if k is odd
if k is even
Suppose S(n, k) is
24
Proof. Consider the sequence (4.1). Since ir7 (S3 ) = Z/2, | S(2, k) I is at most 2. Also, since S4 is
a co-H-space we have for any k, r, (map(S4 , S 4), k) = 7r 5 (S 4 ) = Z/2. A generator of this group is
k + q where k denotes a degree k self map of S4 and + is induced by the comultiplication on S4.
Let f be a self map of HP 2 of degree k and let o(f, f, H) E 7r8 (Hp2 ) denote the obstruction to
homotopy between f and itself relative to the homotopy H on the 4 skeleton. Then one can check
that,
o(f, f,k + 75 ) =r4 oV5 + k[7 4 ,t4]
Since
[74,14]
=
[L4 0 E73,4 0 E13]
=
[t4
4
] 0o-
[4, L4
o
(73 A
7
3
)
= (2v 4 +
Ev') oq7
=0V' 0 77 = 774 0 V
5
is a generator of
7
8
(HP 2 ), we see that the homomorphism
7rmap(S, HP
2
)koH
d7r8 (Hp2)
is surjective if and only if the degree is even, which proves (a).
Since 7r1 2 HP 2 = 7r 11 S 3 = Z/2 there are at most 2 distinct extensions to HP 3 of a self map f of
liHP 2 , which differ precisely by the coaction of the nonzero element E4 E 7r12 HP 2 . By Proposition 4.6
we have
O(E4 -d f) = o(f) ± E4 o 3v 1 2
where k is the degree of f. Now E4 o 3v1 2
p. 69]).
= 64 o v 1 2 ,
while
± k[t 4 , E4]
[t4, 64] = P(E9 ) = E
o 67 (see Toda [To,
Since these two elements generate two distinct Z/2 summands in 7riIHlP 3 we see that the obstruction to extension is always changed by the coaction and hence that d is trivial which proves
(b).
E
We have seen examples in the proof above of how coaction on a map can alter the obstruction to
extension. Therefore we have the following
Remark 4.9. The vanishing of o(f) does not depend exclusively on the degree of
f.
I would like to thank McGibbon for pointing out that this remark also follows from a Theorem of
Mislin [Ms] which states that any self map of HP' of degree 0 is null and the existence of essential
self maps of HP" of degree 0 for n = 3,4 and 5 proved by Marcum and Randall [MR].
In the proof of Proposition 4.8 we saw that o(f) may have a nonzero component along E4 0 V12.
This element of iri 5 HP' is detected in the unstable Adams spectral sequence in filtration 4 so we
have the following
Remark 4.10. o(f) is not necessarily vi-periodic.
Notice however that, in this case, given a self map of HP 3 we can always alter it by coaction so
as to make the obstruction to extension vi-periodic.
Let SHE(X) denote the group of homotopy classes of self homotopy equivalences of the space X.
For n > 2 any element of SHE(HPn) must have degree 1 since there is no self map of HP 2 of degree
-1 (see Theorem 5.2). Hence it follows from the previous proposition that SHE(HP 2 ) = Z/2 and
SHE(HP 3 ) is a group with 4 elements. In fact, Iwase, Maruyama and Oka have shown in [IMO]
that SHE(HP 3 ) = Z/2 e Z/2. In the same paper the authors also show that SHE(HP 4 ) is either 0
or Z/2 and conjecture it is 0.
25
5. HOMOLOGICAL CLASSIFICATION OVER HP".
In this section we discuss the image of the Pontryagin class of quaternionic line bundles over HPn
or, equivalently,the homological classification of self maps of HP".
We begin by describing a conjecture of Feder and Gitler. We then formulate local versions of
the conjecture. We rephrase these in terms of the obstruction to extension of a self map and then
use obstruction theory based on the Adams spectral sequence to construct maps and prove the
conjecture for n < 5. We conclude by formulating the conjecture in terms of geometric dimension
of certain elements in K-theory of HPn.
The Feder-Gitler conjecture. We begin with some preliminaries on the K-theory of HP". Recall
that the coefficient rings for real and complex K-theories are
KO*(pt) = Z[r7, a,,3: 1]/ (
3
, 2a
2
-43)
K*(pt) = Z[v±']
where JrJ = -1, lal = -4,1,31 = -8 and lvi = -2. Recall that there is a natural ring map c induced
by complexification of bundles and a natural KO*-module map r induced by forgetting the complex
structure
c: KO*(X)
- K*(X)
r : K*(X) -
KO*(X)
2
which on coefficients are determined by c(a) = 2v , r(v 2 ) = a.
As usual, we set KO(X) := KO(X),KSp(X) := K0 4 (X),K(X) = K 0 (X).
forgetful map
The natural
KSp(X) -L*K(X)
2
is given by p(x) = v c(X).
Recall also that there are Adams operations ok acting on KO(X) and K(X) which are ring
homomorphisms and are conjugated by the maps c and r.
Using the Atiyah-Hirzebruch spectral sequence it is easy to see that for 1 < n < oc we have
KO*(HP") = KO*[[X4]]I(X4'+1)
K*(HPf) = K*[[X4]]/(X4+1)
We have denoted both generators by x4 because they are identified by the map c. Note that c is an
injection of the torsion free part of KO*(HPf) in K*(EHP') and similarly for r.
HPn
Since the natural projection map Cp2n+1 -+
determines an injection in complex K-theory,
it is easy to find a formula for the action of the Adams operations on K(HPn) = Z[]/xn+l where
we have written x for the ring generator v 2 X4. In particular we have
2
02X = 4x + X
A map f : HP' -+
HP' determines a ring endomorphism of K(HP") commuting with Adams
operations. If we write
f*()
= aix + a2X2 +.
f and the relation 0/2 f*X = f* /, 2 (x) recursively
determines the coefficients an in terms of a 1 . Actually, it is not easy to find an explicit formula for
the coefficients from the recursive relations, but it is proved in [FG] that writing k = a1 ,
then one easily checks that a1 is the degree of
n-1
(5.1)
(k _ i2 )
an= (2)!
(2
i=O
26
Note that since the Adams operations 0I are ring endomorphisms, this gives a formula for the Adams
operations on K(HP')
If we write H for the Hopf bundle over HP', then x = H - 2C hence for any map f, f*x is in
the image of p : KSp(HP') K(H[P'). The image of p is additively generated by the elements
x, 2x 2 ,3,....
This together with (5.1) gives us conditions which must be satisfied by the degrees
of self maps of HP".
Definition 5.1. Let R, be the set of integers satisfying the congruences:
rn-i
(5.2)
(k
Cm :_
-
i2)
i=0
0 mod
(2m)!
(52 (2m)!/2
if m is even
if m is odd
for m = 1,... ,n. Let Rn,, the set of elements in Z(p) satisfying Ci,... , Cn.
The discussion above is now summarized by the following result of [FG]
Theorem 5.2 (Feder and Gitler). If f is a self map of HPn of degree k then k E Rn.
In the same paper, Feder and Gitler made the following
Conjecture 5.3 (Feder and Gitler). If k E Rn there is a self map of HPn of degree k.
The conjecture holds trivially for n = 1. For n = 2 it follows from a result of Arkowitz and
Curjel [AC] and McGibbon [MG2] has given a proof for n = 3. It is also true for n = 00. It is proved
in [FG] that R.. is the set of odd square integers and 0. Sullivan [Su] constructed self maps of these
degrees, the unstable Adams operations, later proved to be unique up to homotopy by Mislin [Ms].
McGibbon [MG1] has given further evidence for the conjecture by proving a stable version which
we will now describe. There is a Hurewicz homomorphism
[EO HP, E-P]-+
2
Theorem 5.4 (McGibbon). (k, k ,...
iff k E Rn.
, k")
End(ft.P)
~Z
is in the image of the stable Hurewicz homomorphism
Thus the maps of Conjecture 5.3 all exist after appropriate suspension. We should note also that
McGibbon obtains in [MG1] the homological classification of stable self maps.
Note that the content of Conjecture 5.3 is that KO-theory together with its primary operations
is sufficiently powerful to detect the homological classification of self maps of JpJ.
This should be
made clear by Proposition 5.8.
In the rest of this section we will gather further evidence for Conjecture 5.3. Our basic strategy
will be to construct maps inductively by showing that the obstruction to extension vanishes. This
will be done using the knowledge of the unstable Adams spectral sequence for HP' from section
2. We will conclude by rephrasing the conjecture in terms of the geometric dimension of certain
elements in K(HP").
The local problem. It follows from Lemma 3.3 that a self map of degree k exists if and only if one
exists after localization at each prime. The local formulations of Theorem 5.2 and Conjecture 5.3
are obtained by replacing Rn with Rn,p. It is therefore helpful to have a manageable description of
Rn,p.
Recall (see [Ei] for example) that k is a 2-adic square and a unit iff it k = 1 mod 8 and, for p an
odd prime, k is a p-adic square iff k = p 2 1u where u reduces to a non-zero square in Z/p and 1 E No.
27
Proposition 5.5. R,, 2 = {k E Z(2) : k = 1 mod 8 or k = 0} and for p odd, Ro,, = {k E Z(p)
k = 12 for some 1 E Z,}. If 2 < n < oo then Rn,p = Ro,, U pep(n)Z(p) where
(i) e 2 (n) = 1 + 2[log 2 n]
for some j C N} for p > 2.
(ii) e,(n) = #{k < n : k = pi or k = (P)p-'
Proof. Let lo, 2 = {k E Z( 2 ) : k = 1 mod 8 or k = 0} and for p odd ncp = {k E Z(p) : k =
12 for some 1 E Zp}. It is easy to check that these sets are contained in the closure of R' =
{0, 1,9, 25,... } in Z(p) with the p-adic topology. Since the solution set of each congruence is closed
we conclude that Ro,,- D A., for every p.
We will now show that R,, = Nt ,, Upep(n)Z(P) which will immediately imply the desired result,
since ep(n) tends to oo with n.
First take p = 2. If k E Z(2) satisfies C 2 then either k = 1 mod 8, that is k E Ro, 2 or k = 2'l for
some 1 E Z(2) so it suffices to consider solutions of the form k = 2eu with u a unit and e > 3. Note
also that because v 2 ((4m + 2)!/2) = v 2 ((4m)!) we can consider only the congruences Cm for m even.
Now assume the result is true for all n < 2 d. Let 2 d < 2m < 2 d+1 and let k = 2eu E R 2 m, 2 with
e > 3 and u E Z). Factoring out units in Z(2) we see that k satisfies C 2m if and only if it satisfies
M-1
2
17 (k - (2i) )
0 mod 22 ((4m)!)
i=O
which, writing k = 41 and noting that v2((4m)!) = 2m + v2 ((2m)!), is equivalent to
M-1
II
m)!)
2
(1- i ) = 0 mod 2v2((2
i=O
So if k satisfies C1,... , C 2m then 1 satisfies C1, ... , Cm. Since 1 is not a unit in Z(2) it follows by
induction that R 2 m, 2 - Rcoo, 2 C 2 2d+1Z( 2 ).
Now we must check that k = 2 2d+1 1 satisfies C 1 , . .. , C2m for any 1 E Z(2). That is we must check
that for each j < m
2j-1
v2
1
(k -
2((4j)!)
2V
i=O
2j-1
2d+1+ Z
2v2(i)
2
2
i+j+...+[j/ 2d]
i=O
2(j +... +[j/2
d])+2d+1-2v2(2j)
2j+j+...+[j/2
d]
j+...+[j/2d]+2d+1
> 2j+2v2(2j)
The last inequality is easy to check if we write 2j = 2'v with v a unit in Z(2) and notice that r < d.
The case when p is odd is similar but simpler. One easily checks that
Rnp =
Z(P)
Rn,p = pZ(p) U Ro,,p
Rp, = p 2 Z(P) U ROP
Notice that k E
if n < (p+ 1)/2
if (p + 1)/2 < n < p
Z(p) satisfies C 1 , C 2 ,. .. , Cn if and only if it satisfies C(p+1)/ 2 , Cp, C 2 , ...
,
[n/
If n > p and k E Rn,, - Nt ,, then we can write k = p 2 v for some v E Z(,). The result now follows
by induction noticing that k satisfies C 1 ,... Cn if and only if v satisfies C 1, ... , C[n/p]
LI
Just as in the integral case, self maps of HP" when localized at a prime are well understood.
The following theorem of Rector (see [Rc]) verifies the local Feder-Gitler conjecture for n = oc.
Theorem 5.6 (Rector). If k E Rc,, there is a self map of HPOO of degree k E
Z(p)
28
Actually only the maps with unit degrees are constructed in [Rc] but the same argument gives
the statement above.
Since self maps of HP give us self maps of HP() for each n, Proposition 5.5 has the following
Corollary 5.7. The Feder-Gitler conjecture holds if there is a self map of
each k such that pep(n) Ik.
HPn of degree k for
The e-invariant of the obstruction. We will now see what restrictions are imposed on the
obstruction to extension of a self map of HPn if its degree is in Rn+1 . Not very surprisingly it
turns out that all one can guarantee is that the component of the obstruction which is detected by
KO-theory and primary operations vanishes.
Let A be the abelian category of abelian groups with Adams operations (see [Ad]). Write KO(X)
for the reduced KO-theory of X and recall that the e-invariant is a group homomorphism
Z -4a ExtA(KO(X),KO(Si+1))
where Z = {a E 7r(X)jKO(a) = 0} and
e(a) = (0
-+
KO(Sj+1 ) -
KO(X U. ej+') -+ KO(X) -
0)
If X is a sphere there is a natural identification of the target of e with a subgroup of Q/Z. The
stable e-invariant of 0 E 7ir(S') is defined by representing / as a class 0' E 7r8m+l Sm for some m
and then setting es(3) = e(#') E Q/Z. This is independent of the choice of m (for all this see [Ad]).
Proposition 5.8. Let
f
be a self map of HP' of degree k and a E 7 4 ,+ 2 S 3 be such that E.(a) =
o(f). Then e-(a) = 0 if and only if k E R,+1.
Proof. Recall from the discussion at the beginning of this section that k E Rn+1 if and only if there
is a ring endomorphism f* of K(HPn+l) commuting with Adams operations and such that f*x is
in the image of the forgetful map p. Clearly this still holds if we replace K by KO and p by r o p.
In turn, this is equivalent to the existence of a map of extensions in the category A of groups with
Adams operations
(5.3)
0
> KO(S4n+4)
> KO(Hpn+l)
0
> KO(S 4 n+ 4 )
>
KO(HPn+1 )
: KO(HP')
: 0
KO(HPn)
> 0
>
Indeed, the elements 0b(r(x)) generate KO(HPn) over the rationals so # is determined by its value
on r(x). In turn, this is determined by f*(r(x)) and the fact that #(r(x)n+l) = kn+lr(X)n+l.
Consider the extension E in A which is (5.3) as an extension of groups but with Adams operations
k = k 2 4,k.
Because KO(HP") is torsion free, the existence of the map of extensions (5.3) is
equivalent to the existence of the corresponding self map of E.
Using the natural identification KO 4 (-) ~ KO(E4 (-)), there is a natural map in A
E
~ (0
KO(S 4 n+ 4 )
-
-
KO( E 4HPn+1 )
-
KO(E4HPn) - 0)
which is termwise an injection of abelian groups. The condition that f*(r(x)) lie in the image of
p o r is now easily seen to be equivalent to the condition that the self map of E extends over this
monomorphism.
We conclude that k E Rn+1 iff the following map of extensions in A exists
0
4 8
> KO(S ,+ )
0
> KO(S n+ )
4
8
>
-
KO(E4HPn+1 )
> KO(E4Pn)
> 0
KO( E 4HPn+1 )
> KO(E4HP")
> 0
29
The existence of the map is equivalent to the following equality in ExtA
e(r 4h) o E4f* = kn+1 o e( E4 h)
where o denotes the Yoneda product in ExtA. Since the e-invariant sends compositions of maps to
Yoneda products and is a group homomorphism these are equivalent to
e(E4(f o h))
=
e(kn+1E4h)
4
=
0
e(E o(f))
HP",
By Proposition 4.3 o(f) factors through the inclusion of the bottom cell of
THJP" Uo(f) e4n+8, we have a map of extensions
0
> KO(S 4 n+ 8 )
>
KO(X)
>
0
> KO(S 4n+ 8 )
=
: 0
KO(E4HJP")
1
letting X
i*
> KO(S 8 L.5, e4 n+ 8 )
KO(S 8 )
-
3'.
0
where i denotes the inclusion of the bottom cell. This is equivalent to the equality
eS(a) o i* = e(o(f))
The long exact sequence in Ext induced by
0
-+
KO(E4HP"/HP1) -+
KO(E4 HP")
-
+
KO(S 8 ) -+
4
0
together with the easily checked fact that HomA(KO(E IHP/HP'), KO(S n+8 )) = 0 show that
composition with i* is injective and this concludes the proof.
E
Remark 5.9. Clearly the proposition also holds locally. That is, for
k, e(o(f)) = 0 if and only if k E Rn+1,p-
4
f
a self map of HP(Ip) of degree
The previous proposition simultaneously makes Conjecture 5.3 seem unlikely and proves it for
n < 3 since the obstruction groups 7r6 S 3 and 7rOS 3 are both cyclic and generated by elements that
suspend to the image of the J-homomorphism (see [To]).
For n > 3 it is no longer true that the obstruction group is detected by the e-invariant so we
can't guarantee that the obstruction to extension vanishes. In fact, cf. Remark 4.9, it doesn't always
vanish. There are self maps of HP3 with degree in R 4 whose obstruction to extension doesn't vanish.
This might cause us to think that the conjecture should fail already for n = 4. That this is not
the case (see Corollary 5.11) is evidence for the conjecture of a stronger nature than had previously
been obtained.
The main theorem. We will now use the Adams spectral sequence obstruction theory to prove
our main theorem. The idea is the same as that of the proof of Proposition 3.6 except that we can
use Proposition 5.8 to rule out the first nontrivial obstruction that occurs in the extension process.
Define
ep(n)
if n < (2p + 1)(p - 1)/2 or n = 3
2n - 3
2n - 5
2
n-(2p+1)q/4+1 3
[4(n - 2)/q]
if
if
if
if
or
p = 2 and n > 3 is even
p=2and n>3isodd
p>3 and (2p+ 1)q/4 n < (2p+ 1)q/4+ [log2 ((2p +1)/3)]
p > 3 and n > ( 2p + 1)q/4 + [log2 ((2p + 1)/3)]
if p = 3 and n > (2p + 1)q/4
Figure 5 shows the graph of 7p(n) at a prime p > 3. It bounds the shaded area over the (t - s)-axis.
30
s
slope 1/q
e,(n)
t - s
FIGURE
5. 7,(n) for p > 3.
Theorem 5.10. For any k E Zip there is a map HPJn -+ HPg of degree kp(n)
Proof. As we have already noted, the e-invariant detects the obstruction group 7r6 S3 and 7 1 OS3 at
the prime 2. By Corollary 2.9, if p is an odd prime and n < (2p + 1)q/4 the obstruction group
7r4n_1fS3
is detected by the e-invariant. This proves the theorem for n < (2p + 1)q/4.
Suppose p is odd. Since p is an exponent for r4n-2S 3 , Proposition 4.6(b) implies that the
obstruction to extension of the composite of self maps of HPn of degree pk and pkI with 1 E Z(p)
vanishes. Therefore if pk is a divisor for HPn with k > 1 then p2k is a divisor for fpn+. Since
ep((2p+1)q/4) = 3, this gives the case of the theorem for (2p+1)q/4 < n < (2p+1)q/4+[log2 ((2p+
1)/3)] and p > 5.
Finally, for the remaining cases, let m = -yp(n) and let k E Z(p). It is enough to produce a map
HPn -+ X (where Xm is as in (2.2)) of degree k and we have such a map for n = 1. We must
show that we can extend this map to HPn. Consider the diagram
XM+1
s41+ 3
hpI
:
m
>
Fm
-PoH
Suppose that an extension to HP' has been constructed and 1 < n. Then, because h has Adams
filtration 1, the obstruction to extending the map to Hpk+1, a : S4k+3 _+ Xm E 7 4 k+3 Xm lifts to
Xm+1.
By Theorem 2.5 there are no nonzero differentials with target Eym+i,4 l+ 3 +m+i for r > 2 and
i > 0. Therefore by Lemma 2.3, a C 7 4 + 3 Xm (1) C w4 1+3 HPP is a class of Adams filtration > m in
7 41+ 3HP' In the range we are considering, either Ep't = 0 or E,' = Z/p corresponds to an element which,
by Theorem 2.5 is detected by the e-invariant. Proposition 5.8 now implies that a is 0 so that we
get a map HP+1 -+ Xm as required to continue the induction.
0
Corollary 5.11. Conjecture 5.3 holds for n < 5.
Proof. This follows immediately from Theorem 5.10 once we note that e 2 (4) = e 2 (5) = 5.
L
Note that the ad hoc nature of the proof of Theorem 5.10 for p > 3 clearly indicates that the
value obtained for 7p(n) is not sharp.
We finish this subsection by pointing out the obstructions that must be overcome in order to
prove the first unsettled case of the Feder-Gitler conjecture at each prime. At the prime 2, the
31
first case that remains unsettled is n = 6. The obstruction in question is the class in dimension 23
detected in filtration 9 in the Adams spectral sequence. This class is vi-periodic and hence should
be possible to analyze. The difficulty resides in finding a way of detecting the class on HP in an
effective way. The means given by Thompson in [Th] requires looping three times.
At an odd prime, the first case that remains unsettled is n = (2p + 1)q/4. The obstruction to be
overcome is the class y in filtration 4 mentioned in the proof of Corollary 2.9. This is not vi-periodic.
Geometric dimension. Since the inclusion
HP'
-+
BSp
classifies an algebra generator X4 of kO* (HPI ), it follows that the map f : HP" -- + HP'
degree k determines by composition with the inclusion HP'
BSp the element
(5.4)
kx 4 +
k(k-1)
2k(k - 1)(k - 4) 3
4! ax 4 +
6!
'3x4
of
E KO (HPn) = KSp(HPn)
+...
Thus the problem of deciding whether there is a self map of HPn of degree k is equivalent to that
of deciding whether the element (5.4) in symplectic K-theory has geometric dimension 1.
Assume that (5.4) has geometric dimension 1 on Hpn- 1 .
: .
BSp(1)
S
7
I,1
I
f
//
HPn-1
BSp(n - 1)
/
.
S 4 -1
/
I
HP"
> BSp(n)
Since [HPn, BSp(n)] = KSp(HP") and KSp(HPn) surjects onto KSp(HPn-1) extensions g of f
always exist and we can consider the problem of lifting g up the tower. There is a sequence of
obstruction sets ok(g) C r4 n-1S 4 (n-k)+3 for 1 < k < n - 1 defined if the previous obstructions
contain 0.
It is not hard to check that under the inclusion map
HP'
-+ BSp(n - k)
the obstruction to extension o(f) is sent to an element of Ok (g) (as long as this obstruction is defined).
Thus we are simply filtering the obstruction group 7r4 n,1HP by
Fk(7 4 l,HP ) = ker(i
4
_1IHP'
-+
ir4n_ 1BSp(k))
and analyzing one filtration quotient at a time. We have
0 = F 1 c F 2 c ... c Fn = 7r4n_ 1 HP
The first obstruction to lifting is just the n-th symplectic Pontryagin class of the symplectic vector
bundle classified by g. It is possible to compute its value and one finds not surprisingly that g can
be chosen so that ol(g) = 0 if and only if the degree of f is in Rn. This gives a different proof of
Theorem 5.2.
Finally note that since 7r4S 0 = 0 it follows that Fn_ 1 = Fn- 2 . Therefore the geometric dimension
of the element (5.4) is < n - 2 if k E Rn. In particular, this gives another proof of Conjecture 5.3
for n < 3.
32
33
6.
CERTAIN SPHERICAL FIBRATIONS.
In this section we consider certain fibrations over HP" with fibre S3. Homologically these look
like quaternionic line bundles so it is tempting to look among them for the bundles whose existence is conjectured in Conjecture 5.3. We will show however, that these fibrations do not contain
representatives of all quaternionic line bundles over HP".
To simplify the statements we will work locally. It is easy to deduce corresponding integral results.
We begin by defining the fibrations mentioned above.
Let 1 be a prime and F1 denote the algebraic closure of the field Fl. According to Friedlander and
Mislin [FM] there exists a map
BSL(2;FI)
-*
BSL(2;C) ~- HP
inducing an isomorphism in homology with Z/n coefficients for (n, 1) = 1. In particular, if p is a
prime not equal to 1 then
H*(BSL(2;Fi); Z /p) = Z/p[x]
with
lxi
= 4. Recall from [FP] for example that
H*(BSL(2; Fin ); Z /p) = Z/p[X]
&Z /p(y)
where lxi = 4, yj = 3 and the generators are connected by a higher Bockstein operation f3 r(y) = x.
Here r is such that the p-Sylow subgroup of SL(2; Fun) is cyclic of order pr for p odd or generalized
quaternionic of order 2 ' for p = 2. By varying 1 and n we can realize any r for p odd and any r > 3
for p = 2. In cohomology, the map determined by the inclusion
BSL(2; Fn) -+ BSL(2; F1)
identifies the classes in degree 4.
Let 7r denote the composite of the two maps above
BSL(2; F1n) -+ HPO
Since the p-completion (see [BK]) BSL(2; Fn )^ is rationally trivial, the p-completion of this map
lifts to HP'. We will also use 7r to denote this lift. A Serre spectral sequence calculation shows
that the homotopy fiber of 7r has the homology of S3,) so we have the following fiber sequence
S 3 {p}
S)
BSL(2; Fin)
+
HP
where S3 {pr } denotes the homotopy fibre of the degree pr self map of S3. In particular, we see (cf.
[Co]) that
QBSL(2; Fin )
~ S31p }
For this reason we will from now on write
BS 3{pr}
BSL(2; Fin )
The fibration
(6.1)
BS 3 {pr} HP
is not homologically distinguishable from a (local) quaternionic line bundle over HPgy with Pontryagin class pr (up to a unit in Z(p)). However we know that for r odd such a bundle can not exist
since upr V ROO, for any u E Z'. Now p' E R,, for certain n so one can hope that the restriction
of (6.1) to HP" has the fibre homotopy type of a quaternionic line bundle. Let
S3p) -4
HPy) -c+ B SHE(S
)
34
be the map classifying (6.1). Then we are asking whether a lift exists in the following diagram
1HTP00
>
HP(n)
c. B SHE(S3p)
()
The following proposition rules this out in general.
Proposition 6.1. If E exists then a extends to HPn+l.
(p)
Proof. Let F, be the homotopy fiber of . Equivalently, F, is the total space of the restriction of
(6.1) to HP. Let Fn,, denote the 4n-skeleton of Fr. Then letting i denote the inclusion, we have
(6.2)
Fn,
>
BS 3{pr}( 4 n)
:
P)
HP-OI
Consider the following diagram
M
4
n+ 3 (pn)
>S 4 ')+ 3
S 4(p)
( n+3
)
(p)
BS31pnj(4n)
II
HP)
H
BS 3{pn}( 4 n+4 )
ffipn+l
(p)
c >
nP
The two left columns are cofiber sequences and the horizontal maps are induced by the map
BS 3 {pr} HP(P). By (6.2), the composition of the two maps in the middle row is null therefore
the dotted arrow exists. This implies that o(a) = E o h is divisible by p.
But by the exponent
theorem [Se], pf is an exponent for 7r4n+ 2 S3) so o(a) = 0 which completes the proof.
El
Thus many of the quaternionic line bundles on HP, do not arise from the fibrations (6.1). For
3
instance, a self map of degree 8 of HP((2)
does not extend to HPf2
and therefore can not be obtained
(2)
by restricting one of the spherical fibrations above. In fact, one can see that this is already the case
for a self map of degree 8 of HP2.
35
REFERENCES
[Ad]
[AC]
[BK]
[BL]
[Co]
[CM]
[Ei]
[FG]
[FP]
[FM]
[Gr]
[HM1]
[HM2]
[IMO]
[Ja]
[Mal]
[Ma2]
[Ma3]
[MR]
[MP]
[MG1]
[MG2]
[MG3]
[Mi]
[MiR]
[Ms]
[MT]
[Rc]
[Sc]
[Se]
[St]
[Su]
[Ta]
[Th]
[To]
[Wh]
J. F. Adams, On the groups J(X).IV, Topology 5 (1966) 21-71.
M. Arkowitz and C. R. Curjel, On maps of H-spaces, Topology 6 (1967) 137-148.
A. K. Bousfield and D. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304,
Springer Verlag (1972).
C. Broto and R. Levi, Loop structures on homotopy fibres of self maps of a sphere, preprint (1998).
F. Cohen, Remarks on the homotopy theory associated to some finite groups, Lecture Notes in Math. 1509
(1990) 95-103.
3
E. Curtis and M. Mahowald, The unstable Adams spectral sequence for S , Contemp. Math. 96, AMS (1989)
125-162.
D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Grad. Texts in Math. 150, Springer
Verlag (1995).
S. Feder and S. Gitler, Mappings of quaternionic projective spaces, Bol. Soc. Mat. Mex. 34 (1975) 12-18.
Z. Fiedorowicz and S. Priddy, Homology of classical groups over finite fields and their associated infinite loop
spaces, Lecture Notes in Math. 674, Springer-Verlag (1978).
E. Friedlander and G. Mislin, Cohomology of classifying spaces of complex Lie groups and related discrete
groups, Comment. Math. Helvetici 59 (1984) 347-361.
B. Gray, On the sphere of origin of infinite families in the homotopy groups of spheres, Topology 8 (1969)
219-232.
J. Harper and H. R. Miller, On the double suspension homomorphism at odd primes, Trans. Amer. Math. Soc.
273 (1982) 319-331.
J. Harper and H. R. Miller, Looping Massey-Peterson towers in: S.M. Salamon, B. Steer, W.A. Sutherland, eds.
Advances in Homotopy Theory(London Math. Soc. Lec. Notes 139, Cambridge University Press, Cambridge,
1989) 69-86.
N. Iwase, K. Maruyama and S. Oka, A note on F(HPn) for n < 4, Math. J. Okayama Univ. 33 (1991) 163-176.
1. M. James, The topology of Stiefel manifolds, London Math. Soc. Lec. Notes 24, Cambridge University Press,
Cambridge (1976).
M. Mahowald, On the double suspension homomorphism, Trans. Amer. Math. Soc. 214 (1975) 169-178.
M. Mahowald, The image of J in the EHP sequence, Annals of Math. 116 (1982) 65-112.
M. Mahowald, The order of the image of the J-homomorphism, Bull. Amer. Math. Soc. 76 (1970) 1310-1313.
H. Marcum and D. Randall, A note on self-mappings of quaternionicprojective spaces, An. Acad. Brasil. Ci.
48 (1976) 7-9.
W. Massey and F. P. Peterson, The mod 2 cohomology structure of certain fiber spaces, Mem. Amer. Math.
Soc. 74 (1967).
C. McGibbon, Self maps of projective spaces, Trans. Amer. Math. Soc. 271 (1982) 325-346.
C. McGibbon, Multiplicative properties of power maps. I, Quart. J. Math. Oxford (2) 31 (1980) 341-350.
C. McGibbon, Multiplicative properties of power maps. II, Trans. Amer. Math. Soc. 274 (1982) 479-508.
H. Miller, A localization theorem in homological algebra, Math. Proc. Camb. Phil. Soc. 84 (1978) 73-84.
H. Miller and D. Ravenel, Mark Mahowald's work on homotopy groups of spheres, Contemp. Math. 146, AMS
(1993) 1-30.
G. Mislin, The homotopy classification of self-maps of infinite quaternionic projective space, Quart. J. Math.
Oxford (2) 38 (1987) 245-257.
R. Mosher and M. Tangora, Cohomology operations and applications in homotopy theory, Harper and Row
Publishers, New York (1968).
3
D. Rector, Loop structures on the homotopy type of S , Lecture Notes in Math. 249 (1971) 99-105.
L. Schwartz, Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture, Chicago
Lectures in Mathematics, University of Chicago Press (1994).
3
P. Selick, Odd primary torsion in 7rk S , Topology 17 (1978) 407-412.
J. Stasheff, H-spaces from a homotopy point of view, Lecture Notes in Math. 161, Springer Verlag (1970).
D. Sullivan, Geometric Topology I. Localization, periodicity and Galois symmetry, MIT Notes (1970).
M. Tangora, Computing the homology of the lambda algebra, Memoirs of the Amer. Math. Soc. 337 (1985).
R. Thompson, The v1-periodic homotopy groups of an unstable sphere at odd primes, Trans. Amer. Math.
Soc. 319 (1990) 535-559.
H. Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies 49, Princeton
University Press (1962).
G. Whitehead, Elements of Homotopy Theory, Grad. Texts in Math 61, Springer Verlag (1978).